can be \(\alpha =\beta =O(1)\)  for \(k\) -means and all its variants, by using the generic algorithm for computing \((\alpha ,\beta )\) -approximation in [1]} with \(\alpha =\beta =O(\log n)\) , and then use the technique for reducing \(\alpha \)  and \(\beta \) . For bounding the approximation of the bi-criteria approximation in [1]} only the pseudo-dimension of the problem is required, which is usually easy to compute as explained below.
